Abstract
In this paper we propose a key exchange protocol (KEP) based on the so-called matrix power function (MPF) defined over a non-commuting platform group. In general, it is not possible to construct KEP using a non-commuting platform group. Therefore we proposed special templates for our public parameters thus allowing us to construct KEP relying on the basic properties of our MPF. Security analysis is based on the decisional Diffie-Hellman (DDH) attack game. We proved that the distribution of the entries of the public session parameter matrices and the shared key matrix asymptotically approaches to uniform with exponential rate. Hence proposed KEP is secure under the DDH assumption. This implies that our protocol is not vulnerable to the computational Diffie-Hellman (CDH) attack. We presented the evidence of CDH security by numerical simulation of linearization attack and showed that it is infeasible.
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References
Álvarez, R., Tortosa, L., Vicent, J., Zamora, A.: A Non-abelian group based on block upper triangular matrices with cryptographic applications. In: Bras-Amorós, M., Høholdt, T. (eds.) AAECC 2009. LNCS, vol. 5527, pp. 117–126. Springer, Heidelberg (2009). https://doi.org/10.1007/978-3-642-02181-7_13
Anshel, I., Anshel, M., Goldfeld, D.: An algebraic method for public-key cryptography. Math. Res. Lett. 6(3), 287–291 (1999)
Boneh, D., Shoup, V.: A graduate course in applied cryptography. 2020. Version 0.5 (2020)
Grundman, H., Smith, T.: Automatic realizability of galois groups of order 16. Proc. Am. Math. Soc. 124(9), 2631–2640 (1996)
Klingler, L.C., Magliveras, S.S., Richman, F., Sramka, M.: Discrete logarithms for finite groups. Computing 85(1–2), 3 (2009)
Ko, K.H., Lee, S.J., Cheon, J.H., Han, J.W., Kang, J., Park, C.: New public-key cryptosystem using braid groups. In: Bellare, M. (ed.) CRYPTO 2000. LNCS, vol. 1880, pp. 166–183. Springer, Heidelberg (2000). https://doi.org/10.1007/3-540-44598-6_10
Lanel, G., Jinasena T., Welihinda B.: A survey of public-key cryptography over non-abelian groups. IJCSNS 21(4), 289 (2021)
Liu, J., Zhang, H., Jia, J.: A linear algebra attack on the non-commuting cryptography class based on matrix power function. In: Chen, K., Lin, D., Yung, M. (eds.) Inscrypt 2016. LNCS, vol. 10143, pp. 343–354. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-54705-3_21
Mihalkovich, A.: On the associativity property of mpf over m16
Mihalkovich, A., Levinskas, M.: Investigation of matrix power asymmetric cipher resistant to linear algebra attack. In: Damaševičius, R., Vasiljevienė, G. (eds.) ICIST 2019. CCIS, vol. 1078, pp. 197–208. Springer, Cham (2019). https://doi.org/10.1007/978-3-030-30275-7_16
Mihalkovich, A., Sakalauskas, E.: Asymmetric cipher based on MPF and its security parameters evaluation. Proc. Lithuanian Math. Soc. Ser. A 53, 72–77 (2012)
Mihalkovich, A., Sakalauskas, E., Venckauskas, A.: New asymmetric cipher based on matrix power function and its implementation in microprocessors efficiency investigation. Elektronika ir Elektrotechnika 19(10), 119–122 (2013)
Patarin, J., Goubin, L.: Trapdoor one-way permutations and multivariate polynomials. In: Han, Y., Okamoto, T., Qing, S. (eds.) ICICS 1997. LNCS, vol. 1334, pp. 356–368. Springer, Heidelberg (1997). https://doi.org/10.1007/BFb0028491
Sakalauskas, E.: Enhanced matrix power function for cryptographic primitive construction. Symmetry 10(2), 43 (2018)
Sakalauskas, E., Luksys, K.: Matrix power function and its application to block cipher s-box construction. Int. J. Inn. Comp., Inf. Contr. 8(4), 2655–2664 (2012)
Sakalauskas, E., Mihalkovich, A.: New asymmetric cipher of non-commuting cryptography class based on matrix power function. Informatica 25(2), 283–298 (2014)
Sakalauskas, E., Mihalkovich, A.: Improved asymmetric cipher based on matrix power function resistant to linear algebra attack. Informatica 28(3), 517–524 (2017)
Sakalauskas, E., Mihalkovich, A.: MPF problem over modified medial semigroup is np-complete. Symmetry 10(11), 571 (2018)
Sakalauskas, E., Listopadskis, N., Tvarijonas, P.: Key agreement protocol (KAP) based on matrix power function (2008)
Shpilrain, V., Ushakov, A.: The conjugacy search problem in public key cryptography: unnecessary and insufficient. Appl. Algebra Eng. Commun. Comput. 17(3–4), 285–289 (2006)
Stickel, E.: A new public-key cryptosystem in non abelian groups. Proceedings of the Thirteenth International Conference on Information Systems Development. Vilnius Technika: 70–80, Vilnius (2004)
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Mihalkovich, A., Sakalauskas, E., Levinskas, M. (2022). Key Exchange Protocol Based on the Matrix Power Function Defined Over 
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In: Arai, K. (eds) Intelligent Computing. SAI 2022. Lecture Notes in Networks and Systems, vol 508. Springer, Cham. https://doi.org/10.1007/978-3-031-10467-1_32
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